Automorphism groups of pure integral octonions

A matrix representation of the automorphism group of pure integral octonions constituting the root system of E₇ is constructed. It is shown that it is a finite subgroup of the exceptional group of G₂ of order 12096, called the adjoint Chevalley group G₂(2). Its four maximal subgroups of orders 432, 192, 192' and 336 preserve, respectively, the octonionic root systems of E₆, SO(12), SU(2)³*SO(8) and SU(8). It is also shown explicitly that the full automorphism group of the pure octonions +or-e_i (i=1,..., 7) constituting the roots of SU(2)⁷ is a group of order 1344. Possible implications in physics are discussed.